# Connectedness of the set having a fixed distance from a closed set 2

This question is related to this one: Connectedness of the set having a fixed distance from a closed set. Suppose $$F$$ is a closed and connected set in $$mathbb{R}^n$$ ($$n>1$$). Suppose the complement of $$F$$ is connected and let
$$A={xin mathbb{R}^n: dist(x, F)=delta},$$ where $$delta>0$$ is fixed, and $$dist$$ is the Euclidean distance. If $$F$$ is unbounded with empty interior, can the complement of $$A$$ still have a bounded component?

MathOverflow Asked by M. Rahmat on January 23, 2021

Here is a picture of the set $$F$$ (in red) in my comment above (the black lines represent the sphere of radius 2). There is a component of $$A$$ inside the sphere and a component outside the sphere.

Correct answer by Anthony Quas on January 23, 2021

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